This morning I saw a commercial where a man cheered "more than half of our mutual funds outperforms the market's average!" I can't help thinking about this statement statistically.
It will be an absolutely dumb statement if it was not "average" but "median". For "average", if the distribution of the returns is highly right-skewed, with a more than 50% probability to perform higher than the mean is a good indicator. But can the copy-writer of that commercial be so statistically sophiscated? This could be an amusing w1111 example in the future. Even though mutual funds may not be the most appropriate subject. I used to have students who complained about the car examples I used since they had little experience with automobiles.
This reminded me of my conversation with Ying Wei the past Monday. It was on the loss of efficience when estimating mean of a Gaussian distribution using median compared with using mean. That reminded me of the recent popular show on NBC: Deal or No Deal (DoND).
Here is my version of what that show is about: the show starts with 26 closed cases contain 26 fixed money values range from $0.01 to $1,000,000. The contestant will open several cases randomly in batches. The cases opened are eliminated from the board (i.e., can not be won by the contestant). After each batch of cases opened, the show will pause. Looking at the remained undiscovered money amounts, a banker will offer the contestant an amount of money to make him/her stop, (from winning the biggest remaining value, of course). If the contestant refuses the offer, he/she will have to eliminate one or more amounts by random guessing, which will actually make the next offer drop.
From the contestant stand point, he/she should accept offer that is higher than the MEDIAN since he/she only play once. If he/she keeps on playing, there is a 50-50 chance that he/she leaves with value lower than the offer. On the banker side, he needs to make offer that is much lower than the mean since he needs to play the game many times. Thus, it is not a surprise to me that every time the bank makes an offer, it is always much lower than the mean of the remaining values. I still yet to figure out the magical amount (offer-median) and the reasoning behind it.